Homological growth of Artin kernels in positive characteristic

TL;DR

This paper establishes an analogue of the L"uck Approximation Theorem in positive characteristic for certain groups, linking mod p homology growth to group homology with universal coefficients, with applications in topology and geometric group theory.

Contribution

It introduces a positive characteristic version of the L"uck Approximation Theorem for RFRS groups, including right-angled Artin and Bestvina--Brady groups, and explores related inequalities and applications.

Findings

Mod p homology growth equals the dimension of group homology with universal division ring coefficients.

The invariants are independent of the residual chain for certain groups.

Applications include results on fibring, amenable category, and minimal volume entropy.

Abstract

We prove an analogue of the L\"uck Approximation Theorem in positive characteristic for certain residually finite rationally soluble (RFRS) groups including right-angled Artin groups and Bestvina--Brady groups. Specifically, we prove that the mod $p$ homology growth equals the dimension of the group homology with coefficients in a certain universal division ring and this is independent of the choice of residual chain. For general RFRS groups we obtain an inequality between the invariants. We also consider a number of applications to fibring, amenable category, and minimal volume entropy.